This work presents an adaptive phase field framework for the simulation of crack nucleation and propagation in brittle materials subject to low-cycle loading. We introduce a fatigue history strain parameter within the phase-field framework to capture the fatigue effect. The resulting coupled differential equations are solved using a staggered iteration scheme. In order to improve the computational efficiency of the phase-field model, an adaptive refinement scheme is introduced. This scheme utilizes an artificial threshold that takes into consideration of both phase-field variables and cumulative fatigue history field variables to determine the elements to be refined. The handling of the hanging nodes resulting from mesh refinement is accomplished through the application of the variable-node element technique, offering a flexible and efficient mesh integration technique. We validate our proposed numerical scheme through five representative numerical examples: single-edge cracked specimen, compact tension specimen, double-edge cracked panel, plate with a hole and plate with multi-cracks and hole. The results demonstrate that the adaptive mesh refinement scheme significantly reduces computational costs without compromising the accuracy of the numerical predictions.

中文翻译：

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脆性材料中的低周疲劳裂纹扩展：使用可变节点单元的自适应相场建模

这项工作提出了一种自适应相场框架，用于模拟低周载荷下脆性材料中的裂纹形核和扩展。我们在相场框架内引入疲劳历史应变参数来捕获疲劳效应。使用交错迭代方案求解所得耦合微分方程。为了提高相场模型的计算效率，引入了自适应细化方案。该方案利用人工阈值，考虑相场变量和累积疲劳历史场变量来确定要细化的元素。通过应用变节点单元技术完成网格细化产生的悬挂节点的处理，提供了灵活高效的网格集成技术。我们通过五个代表性的数值例子验证了我们提出的数值方案：单边裂纹试件、紧凑拉伸试件、双边裂纹面板、带孔板和多裂纹和孔板。结果表明，自适应网格细化方案显着降低了计算成本，而不会影响数值预测的准确性。